Richard J. Pike

Elevation-Relief Ratio, Hypsometric Integral, and Geomorphic Area-Altitude Analysis

Mathematical proof establishes identity of hypsometric integral and elevation-relief ratio, two quantitative topographic descriptors developed independently of one another for entirely different purposes. Operationally, values of both measures are in excellent agreement for arbitrarily bounded topographic samples, as well as for low-order fluvial watersheds. By using a point-sampling technique rather than planimetry, elevation-relief ratio (defined as mean elevation minus minimum elevation divided by relief) is calculated manually in about a third of the time required for the hypsometric integral.

The geometric signature: quantifying landslide-terrain types from digital elevation models

Topography of various types and scales can be fingerprinted by computer analysis of altitude matrices (digital elevation models, or DEMs). The critical analytic tool is the geometric signature, a set of measures that describes topographic form well enough to distinguish among geomorphically disparate landscapes. Different surficial processes create topography with diagnostic forms that are recognizable in the field. The geometric signature abstracts those forms from contour maps or their DEMs and expresses them numerically. This multivariate characterization enables once-in-tractable problems to be addressed. The measures that constitute a geometric signature express different but complementary attributes of topographic form. Most parameters used here are statistical estimates of central tendency and dispersion for five major categories of terrain geometry; altitude, altitude variance spectrum, slope between slope reversals, and slope and its curvature at fixed slope lengths. As an experimental application of geometric signatures, two mapped terrain types associated with different processes of shallow landsliding in Marin County, California, were distinguished consistently by a 17-variable description of topography from 21×21 DEMs (30-m grid spacing). The small matrix is a statistical window that can be used to scan large DEMs by computer, thus potentially automating the mapping of contrasting terrain types. The two types in Marin County host either (1) slow slides: earth flows and slump-earth flows, or (2) rapid flows: debris avalanches and debris flows. The signature approach should adapt to terrain taxonomy and mapping in other areas, where conditions differ from those in Central California.

Formation of complex impact craters: Evidence from Mars and other planets

The simple-to-complex transition for impact craters on Mars occurs at diameters between about 3 and 8 km. Ballistically emplaced ejecta surround primarily those craters that have a simple interior morphology, whereas ejecta displaying features attributable to fluid flow are mostly restricted to complex craters. Size-dependent characteristics of 73 relatively fresh Martian craters, emphasizing the new depth/diameter (d/D) data of D. W. G. Arthur (1980, to be submitted for publication), test two hypotheses for the mode of formation of central peaks in complex craters. In particular, five features appear sequentially with increasing crater size: first flat floors (3–4 km), then central peaks and shallower depths (4–5 km), next scalloped rims (∂ km), and lastly terraced walls (∼8 km). This relative order indicates that a shallow depth of excavation and an unspecified rebound mechanism, not centripetal collapse and deep sliding, have produced central peaks and in turn have facilitated failure of the rim. The mechanism of formation of a shallow crater remains elusive, but probably operates only at the excavation stage of impact. This interpretation is consistent with two separate and complementary lines of evidence. First, field data have documented only shallow subsurface deformation and a shallow transient cavity in complex terrestrial meteorite craters and in certain surface-burst explosion craters; thus the shallow transient cavities of complex craters never were geometrically similar to the deep cavities of simple craters. Second, the average depths of complex craters and the diameters marking the transition from simple to complex craters on Mars and on three other terrestrial planets vary inversely with gravitational acceleration at the planetary surface, g, a variable more important in the excavation of a crater than in any subsequent modification of its geometry. The new interpretation is summarized diagrammatically for complex craters on all planets.

Basin-ring spacing on the Moon, Mercury and Mars

Radial spacing between concentric rings of impact basins that lack central peaks is statistically similar and nonrandom on the Moon, Mercury, and Mars, both inside and outside the main ring. One spacing interval, (2.0 ± 0.3)0.5D, or an integer multiple of it, dominates most basin rings. Three analytical approaches yield similar results from 296 remapped or newly mapped rings of 67 multi-ringed basins: least-squares of rank-grouped rings, least-squares of rank and ring diameter for each basin, and averaged ratios of adjacent rings. Analysis of 106 rings of 53 two-ring basins by the first and third methods yields an integer multiple (2 ×) of 2.00.5D. There are two exceptions: (1) Rings adjacent to the main ring of multi-ring basins are consistently spaced at a slightly, but significantly, larger interval, (2.1 ± 0.3)0.5D; (2) The 88 rings of 44 protobasins (large peak-plus-inner-ring craters) are spaced at an entirely different interval (3.3 ± 0.6)0.5D.The statistically constant and target-invariant spacing of so many rings suggests that this characteristic may constrain formational models of impact basins on the terrestrial planets. The key elements of such a constraint include: (1) ring positions may not have been located by the same process(es) that formed ring topography; (2) ring location and emplacement of ring topography need not be coeval; (3) ring location, but not necessarily the mode of ring emplacement, reflects one process that operated at the time of impact; and (4) the process yields similarly-disposed topographic features that are spatially discrete at 20.5D intervals, or some multiple, rather than continuous. These four elements suggest that some type of wave mechanism dominates the location, but not necessarily the formation, of basin rings. The waves may be standing, rather than travelling. The ring topography itself may be emplaced at impact by this and/or other mechanisms and may reflect additional, including post-impact, influences.

Ejecta from large craters on the moon - Comments on the geometric model of McGetchin et al

Abstract Amendments to a quantitative scheme developed by T.R. McGetchin et al. (1973) for predicting the distribution of ejecta from lunar basins yield substantially thicker estimates of ejecta, deposited at the basin rim-crest and at varying ranges byond, than does the original model. Estimates of the total volume of material ejected from a basin, illustrated by Imbrium, also are much greater. Because many uncertainties affect any geometric model developed primarily from terrestrial analogs of lunar craters, predictions of ejecta thickness and volume on the Moon may range within at least an order of magnitude. These problems are exemplified by the variability of T , thickness of ejecta at the rim-crest of terrestrial experimental craters. The proportion of T to crater rim-height depends critically upon scaled depth-of-burst and whether the explosive is nuclear or chemical.

Crater dimensions from apollo data and supplemental sources

A catalog of crater dimensions that were compiled mostly from the new Apollo-based Lunar Topographic Orthophotomaps is presented in its entirety. Values of crater diameter, depth, rim height, flank width, circularity, and floor diameter (where applicable) are tabulated for a sample of 484 craters on the Moon and 22 craters on Earth. Systematic techniques of mensuration are detailed. The lunar craters range in size from 400 m to 300 km across and include primary impact craters of the main sequence, secondary impact craters, craterlets atop domes and cones, and dark-halo craters. The terrestrial craters are between 10 m and 22.5 km in diameter and were formed by meteorite impact.

Genetic implications of the shapes of martian and lunar craters

Abstract Craters on Mars and the Moon are alike in that larger craters differ in shape from smaller ones, and older craters differ in shape from younger ones. Smoothed depth-diameter curves for 41 large martian craters photographed by Mariner IV inflect at a crater diameter of 10–20km in a manner similar to curves for lunar craters. Below 10–20km, both depth-diameter curves are linear with a slope of roughly 1.0; above this threshold range, the curves assume a much lower slope. Diminution of lunar crater depth-diameter ratios with age indicates that the shapes of lunar and, by inference, martian craters have changed systematically since formation. Martian craters sampled here are shallower than most pre-Imbrian lunar craters. By analogy with the Moon, martian craters seem both to vary in initial shape according to the energy of the impact that formed them and to have been modified subsequently by endogenic and surface processes. A proposed model for the geologic development of large martian and lunar craters outlines a time- dependent sequence of events. Craters which have undergone rapid isostatic adjustment on the Moon have distinctive morphologies and occur preferentially along mare basin-upland margins.

Craters on Earth, Moon, and Mars: Multivariate classification and mode of origin

Abstract Testing extraterrestrial craters and candidate terrestrial analogs for morphologic similitude is treated as a problem in numerical taxonomy. According to a principal-components solution and a cluster analysis, 402 representative craters on the Earth, the Moon, and Mars divide into two major classes of contrasting shapes and modes of origin. Craters of net accumulation of material (cratered lunar domes, Martian “calderas,” and all terrestrial volcanoes except maars and tuff rings) group apart from craters of excavation (terrestrial meteorite impact and experimental explosion craters, typical Martian craters, and all other lunar craters). Maars and tuff rings belong to neither group but are transitional. The classification criteria are four independent attributes of topographic geometry derived from seven descriptive variables by the principal-components transformation. Morphometric differences between crater bowl and raised rim constitute the strongest of the four components. Although single topographic variables cannot confidently predict the genesis of individual extraterrestrial craters, multivariate statistical models constructed from several variables can distinguish consistently between large impact craters and volcanoes.

Digital Terrain Modeling and Industrial Surface Metrology: Converging Realms

Digital terrain modeling has a micro- and nanoscale counterpart in surface metrology, the numerical characterization of industrial surfaces. Instrumentation in semiconductor manufacturing and other high-technology fields can now contour surface irregularities down to the atomic scale. Surface metrology has been revolutionized by its ability to manipulate square-grid height matrices that are analogous to the digital elevation models (DEMs) used in physical geography. Because the shaping of industrial surfaces is a spatial process, the same concepts of analytical cartography that represent ground-surface form in geography evolved independently in metrology. The surface topography of manufactured components, exemplified here by automobile-engine cylinders, is routinely modeled by variogram analysis, relief shading, and most other techniques of parameterization and visualization familiar to geography. This article introduces industrial surface-metrology, examines the field in the context of terrain modeling and geomorphology and notes their similarities and differences, and raises theoretical issues to be addressed in progressing toward a unified practice of surface morphometry.

Nano-metrology and terrain modelling - convergent practice in surface characterisation

Abstract The quantification of magnetic-tape and disk topography has a macro-scale counterpart in the Earth sciences — terrain modelling , the numerical representation of relief and pattern of the ground surface. The two practices arose independently and continue to function separately. This methodological paper introduces terrain modelling, discusses its similarities to and differences from industrial surface metrology, and raises the possibility of a unified discipline of quantitative surface characterisation. A brief discussion of an Earth–science problem, subdividing a heterogeneous terrain surface from a set of sample measurements, exemplifies a multivariate statistical procedure that may transfer to tribological applications of 3–D metrological height data.